\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  6.0 & +  3.00 x_{1} & -2.00 x_{2} & -5.00 x_{3} & -3.00 x_{4} & -7.00 x_{5} & -7.00 x_{6} & + 10.00 x_{7} &   \\
 x_{10}   &  18.0 & -9.00 x_{1} & +  5.00 x_{2} & +  7.00 x_{3} &   & -6.00 x_{5} &   & +  2.00 x_{7} & -4.00 x_{8}\\
 x_{11}   &  -8.0 & +  6.00 x_{1} &   & +  3.00 x_{3} & -2.00 x_{4} & +  4.00 x_{5} & +  3.00 x_{6} & -3.00 x_{7} & -1.00 x_{8}\\
 x_{12}   &  54.0 & +  9.00 x_{1} & -6.00 x_{2} & -7.00 x_{3} & -9.00 x_{4} & -9.00 x_{5} & +  6.00 x_{6} & -9.00 x_{7} & +  7.00 x_{8}\\
 x_{13}   &  21.0  &   & -1.00 x_{2} & -7.00 x_{3} & +  1.00 x_{4} & -4.00 x_{5} & +  2.00 x_{6} & -4.00 x_{7} & +  4.00 x_{8}\\
\hline
z    &  0.0 & +  4.00 x_{1} & -4.00 x_{2} & -4.00 x_{3} & +  2.00 x_{4} & +  3.00 x_{5} & +  5.00 x_{6} & +  3.00 x_{7} & +  5.00 x_{8}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{8}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -3.00 y_{9} & +  9.00 y_{10} & -6.00 y_{11} & -9.00 y_{12} &   \\
 y_{2}   &  1.0 & +  2.00 y_{9} & -5.00 y_{10} &   & +  6.00 y_{12} & +  1.00 y_{13}\\
 y_{3}   &  1.0 & +  5.00 y_{9} & -7.00 y_{10} & -3.00 y_{11} & +  7.00 y_{12} & +  7.00 y_{13}\\
 y_{4}   &  1.0 & +  3.00 y_{9} &   & +  2.00 y_{11} & +  9.00 y_{12} & -1.00 y_{13}\\
 y_{5}   &  1.0 & +  7.00 y_{9} & +  6.00 y_{10} & -4.00 y_{11} & +  9.00 y_{12} & +  4.00 y_{13}\\
 y_{6}   &  1.0 & +  7.00 y_{9} &   & -3.00 y_{11} & -6.00 y_{12} & -2.00 y_{13}\\
 y_{7}   &  1.0 & -10.00 y_{9} & -2.00 y_{10} & +  3.00 y_{11} & +  9.00 y_{12} & +  4.00 y_{13}\\
 y_{8}   &  1.0  &   & +  4.00 y_{10} & +  1.00 y_{11} & -7.00 y_{12} & -4.00 y_{13}\\
\hline
z    &  -0 & -6.00 y_{9} & -18.00 y_{10} & +  8.00 y_{11} & -54.00 y_{12} & -21.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 2 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{11}   &  0.166666666667 & -0.50 y_{9} & +  1.50 y_{10} & -0.17 y_{1} & -1.50 y_{12} &   \\
 y_{2}   &  1.0 & +  2.00 y_{9} & -5.00 y_{10} &   & +  6.00 y_{12} & +  1.00 y_{13}\\
 y_{3}   &  0.5 & +  6.50 y_{9} & -11.50 y_{10} & +  0.50 y_{1} & + 11.50 y_{12} & +  7.00 y_{13}\\
 y_{4}   &  1.33333333333 & +  2.00 y_{9} & +  3.00 y_{10} & -0.33 y_{1} & +  6.00 y_{12} & -1.00 y_{13}\\
 y_{5}   &  0.333333333333 & +  9.00 y_{9} &   & +  0.67 y_{1} & + 15.00 y_{12} & +  4.00 y_{13}\\
 y_{6}   &  0.5 & +  8.50 y_{9} & -4.50 y_{10} & +  0.50 y_{1} & -1.50 y_{12} & -2.00 y_{13}\\
 y_{7}   &  1.5 & -11.50 y_{9} & +  2.50 y_{10} & -0.50 y_{1} & +  4.50 y_{12} & +  4.00 y_{13}\\
 y_{8}   &  1.16666666667 & -0.50 y_{9} & +  5.50 y_{10} & -0.17 y_{1} & -8.50 y_{12} & -4.00 y_{13}\\
\hline
z    &  1.33333333333 & -10.00 y_{9} & -6.00 y_{10} & -1.33 y_{1} & -66.00 y_{12} & -21.00 y_{13}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  10.0 & +  0.50 x_{11} & -2.00 x_{2} & -6.50 x_{3} & -2.00 x_{4} & -9.00 x_{5} & -8.50 x_{6} & + 11.50 x_{7} & +  0.50 x_{8}\\
 x_{10}   &  6.0 & -1.50 x_{11} & +  5.00 x_{2} & + 11.50 x_{3} & -3.00 x_{4} &   & +  4.50 x_{6} & -2.50 x_{7} & -5.50 x_{8}\\
 x_{1}   &  1.33333333333 & +  0.17 x_{11} &   & -0.50 x_{3} & +  0.33 x_{4} & -0.67 x_{5} & -0.50 x_{6} & +  0.50 x_{7} & +  0.17 x_{8}\\
 x_{12}   &  66.0 & +  1.50 x_{11} & -6.00 x_{2} & -11.50 x_{3} & -6.00 x_{4} & -15.00 x_{5} & +  1.50 x_{6} & -4.50 x_{7} & +  8.50 x_{8}\\
 x_{13}   &  21.0  &   & -1.00 x_{2} & -7.00 x_{3} & +  1.00 x_{4} & -4.00 x_{5} & +  2.00 x_{6} & -4.00 x_{7} & +  4.00 x_{8}\\
\hline
z    &  -1.33333333333 & -0.17 x_{11} & -1.00 x_{2} & -0.50 x_{3} & -1.33 x_{4} & -0.33 x_{5} & -0.50 x_{6} & -1.50 x_{7} & -1.17 x_{8}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  10.0 & +  0.50 x_{11} & -2.00 x_{2} & -6.50 x_{3} & -2.00 x_{4} & -9.00 x_{5} & -8.50 x_{6} & + 11.50 x_{7} & +  0.50 x_{8}\\
 x_{10}   &  6.0 & -1.50 x_{11} & +  5.00 x_{2} & + 11.50 x_{3} & -3.00 x_{4} &   & +  4.50 x_{6} & -2.50 x_{7} & -5.50 x_{8}\\
 x_{1}   &  1.33333333333 & +  0.17 x_{11} &   & -0.50 x_{3} & +  0.33 x_{4} & -0.67 x_{5} & -0.50 x_{6} & +  0.50 x_{7} & +  0.17 x_{8}\\
 x_{12}   &  66.0 & +  1.50 x_{11} & -6.00 x_{2} & -11.50 x_{3} & -6.00 x_{4} & -15.00 x_{5} & +  1.50 x_{6} & -4.50 x_{7} & +  8.50 x_{8}\\
 x_{13}   &  21.0  &   & -1.00 x_{2} & -7.00 x_{3} & +  1.00 x_{4} & -4.00 x_{5} & +  2.00 x_{6} & -4.00 x_{7} & +  4.00 x_{8}\\
\hline
z    &  5.33333333333 & +  0.67 x_{11} & -4.00 x_{2} & -6.00 x_{3} & +  3.33 x_{4} & +  0.33 x_{5} & +  3.00 x_{6} & +  5.00 x_{7} & +  5.67 x_{8}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  10.0 & +  0.50 x_{11} & -2.00 x_{2} & -6.50 x_{3} & -2.00 x_{4} & -9.00 x_{5} & -8.50 x_{6} & + 11.50 x_{7} & +  0.50 x_{8}\\
 x_{10}   &  6.0 & -1.50 x_{11} & +  5.00 x_{2} & + 11.50 x_{3} & -3.00 x_{4} &   & +  4.50 x_{6} & -2.50 x_{7} & -5.50 x_{8}\\
 x_{1}   &  1.33333333333 & +  0.17 x_{11} &   & -0.50 x_{3} & +  0.33 x_{4} & -0.67 x_{5} & -0.50 x_{6} & +  0.50 x_{7} & +  0.17 x_{8}\\
 x_{12}   &  66.0 & +  1.50 x_{11} & -6.00 x_{2} & -11.50 x_{3} & -6.00 x_{4} & -15.00 x_{5} & +  1.50 x_{6} & -4.50 x_{7} & +  8.50 x_{8}\\
 x_{13}   &  21.0  &   & -1.00 x_{2} & -7.00 x_{3} & +  1.00 x_{4} & -4.00 x_{5} & +  2.00 x_{6} & -4.00 x_{7} & +  4.00 x_{8}\\
\hline
z    &  5.33333333333 & +  0.67 x_{11} & -4.00 x_{2} & -6.00 x_{3} & +  3.33 x_{4} & +  0.33 x_{5} & +  3.00 x_{6} & +  5.00 x_{7} & +  5.67 x_{8}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{9}   &  6.0 & +  1.50 x_{11} & -5.33 x_{2} & -14.17 x_{3} & +  0.67 x_{10} & -9.00 x_{5} & -11.50 x_{6} & + 13.17 x_{7} & +  4.17 x_{8}\\
 x_{4}   &  2.0 & -0.50 x_{11} & +  1.67 x_{2} & +  3.83 x_{3} & -0.33 x_{10} &   & +  1.50 x_{6} & -0.83 x_{7} & -1.83 x_{8}\\
 x_{1}   &  2.0  &   & +  0.56 x_{2} & +  0.78 x_{3} & -0.11 x_{10} & -0.67 x_{5} &   & +  0.22 x_{7} & -0.44 x_{8}\\
 x_{12}   &  54.0 & +  4.50 x_{11} & -16.00 x_{2} & -34.50 x_{3} & +  2.00 x_{10} & -15.00 x_{5} & -7.50 x_{6} & +  0.50 x_{7} & + 19.50 x_{8}\\
 x_{13}   &  23.0 & -0.50 x_{11} & +  0.67 x_{2} & -3.17 x_{3} & -0.33 x_{10} & -4.00 x_{5} & +  3.50 x_{6} & -4.83 x_{7} & +  2.17 x_{8}\\
\hline
z    &  12.0 & -1.00 x_{11} & +  1.56 x_{2} & +  6.78 x_{3} & -1.11 x_{10} & +  0.33 x_{5} & +  8.00 x_{6} & +  2.22 x_{7} & -0.44 x_{8}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{2}   &  1.125 & +  0.28 x_{11} & -0.19 x_{9} & -2.66 x_{3} & +  0.12 x_{10} & -1.69 x_{5} & -2.16 x_{6} & +  2.47 x_{7} & +  0.78 x_{8}\\
 x_{4}   &  3.875 & -0.03 x_{11} & -0.31 x_{9} & -0.59 x_{3} & -0.12 x_{10} & -2.81 x_{5} & -2.09 x_{6} & +  3.28 x_{7} & -0.53 x_{8}\\
 x_{1}   &  2.625 & +  0.16 x_{11} & -0.10 x_{9} & -0.70 x_{3} & -0.04 x_{10} & -1.60 x_{5} & -1.20 x_{6} & +  1.59 x_{7} & -0.01 x_{8}\\
 x_{12}   &  36.0 & +  0.00 x_{11} & +  3.00 x_{9} & +  8.00 x_{3} & +  0.00 x_{10} & + 12.00 x_{5} & + 27.00 x_{6} & -39.00 x_{7} & +  7.00 x_{8}\\
 x_{13}   &  23.75 & -0.31 x_{11} & -0.12 x_{9} & -4.94 x_{3} & -0.25 x_{10} & -5.12 x_{5} & +  2.06 x_{6} & -3.19 x_{7} & +  2.69 x_{8}\\
\hline
z    &  13.75 & -0.56 x_{11} & -0.29 x_{9} & +  2.65 x_{3} & -0.92 x_{10} & -2.29 x_{5} & +  4.65 x_{6} & +  6.06 x_{7} & +  0.77 x_{8}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{3}   &  0.423529411765 & +  0.11 x_{11} & -0.07 x_{9} & -0.38 x_{2} & +  0.05 x_{10} & -0.64 x_{5} & -0.81 x_{6} & +  0.93 x_{7} & +  0.29 x_{8}\\
 x_{4}   &  3.62352941176 & -0.09 x_{11} & -0.27 x_{9} & +  0.22 x_{2} & -0.15 x_{10} & -2.44 x_{5} & -1.61 x_{6} & +  2.73 x_{7} & -0.71 x_{8}\\
 x_{1}   &  2.32941176471 & +  0.08 x_{11} & -0.05 x_{9} & +  0.26 x_{2} & -0.07 x_{10} & -1.16 x_{5} & -0.63 x_{6} & +  0.95 x_{7} & -0.22 x_{8}\\
 x_{12}   &  39.3882352941 & +  0.85 x_{11} & +  2.44 x_{9} & -3.01 x_{2} & +  0.38 x_{10} & +  6.92 x_{5} & + 20.51 x_{6} & -31.56 x_{7} & +  9.35 x_{8}\\
 x_{13}   &  21.6588235294 & -0.84 x_{11} & +  0.22 x_{9} & +  1.86 x_{2} & -0.48 x_{10} & -1.99 x_{5} & +  6.07 x_{6} & -7.78 x_{7} & +  1.24 x_{8}\\
\hline
z    &  14.8705882353 & -0.28 x_{11} & -0.48 x_{9} & -1.00 x_{2} & -0.79 x_{10} & -3.97 x_{5} & +  2.50 x_{6} & +  8.52 x_{7} & +  1.55 x_{8}\\
\end{array}\]


 $ x_{6} $ enters and $ x_{3} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.521739130435 & +  0.13 x_{11} & -0.09 x_{9} & -0.46 x_{2} & +  0.06 x_{10} & -0.78 x_{5} & -1.23 x_{3} & +  1.14 x_{7} & +  0.36 x_{8}\\
 x_{4}   &  2.78260869565 & -0.30 x_{11} & -0.13 x_{9} & +  0.97 x_{2} & -0.25 x_{10} & -1.17 x_{5} & +  1.99 x_{3} & +  0.88 x_{7} & -1.29 x_{8}\\
 x_{1}   &  2.0 & -0.00 x_{11} & +  0.00 x_{9} & +  0.56 x_{2} & -0.11 x_{10} & -0.67 x_{5} & +  0.78 x_{3} & +  0.22 x_{7} & -0.44 x_{8}\\
 x_{12}   &  50.0869565217 & +  3.52 x_{11} & +  0.65 x_{9} & -12.52 x_{2} & +  1.57 x_{10} & -9.13 x_{5} & -25.26 x_{3} & -8.09 x_{7} & + 16.78 x_{8}\\
 x_{13}   &  24.8260869565 & -0.04 x_{11} & -0.30 x_{9} & -0.96 x_{2} & -0.13 x_{10} & -6.74 x_{5} & -7.48 x_{3} & -0.83 x_{7} & +  3.43 x_{8}\\
\hline
z    &  16.1739130435 & +  0.04 x_{11} & -0.70 x_{9} & -2.15 x_{2} & -0.65 x_{10} & -5.93 x_{5} & -3.08 x_{3} & + 11.38 x_{7} & +  2.45 x_{8}\\
\end{array}\]


 $ x_{7} $ enters and $ x_{12} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  7.61290322581 & +  0.63 x_{11} & +  0.01 x_{9} & -2.24 x_{2} & +  0.28 x_{10} & -2.08 x_{5} & -4.81 x_{3} & -0.14 x_{12} & +  2.74 x_{8}\\
 x_{4}   &  8.25806451613 & +  0.08 x_{11} & -0.06 x_{9} & -0.40 x_{2} & -0.08 x_{10} & -2.17 x_{5} & -0.78 x_{3} & -0.11 x_{12} & +  0.54 x_{8}\\
 x_{1}   &  3.37634408602 & +  0.10 x_{11} & +  0.02 x_{9} & +  0.21 x_{2} & -0.07 x_{10} & -0.92 x_{5} & +  0.08 x_{3} & -0.03 x_{12} & +  0.02 x_{8}\\
 x_{7}   &  6.1935483871 & +  0.44 x_{11} & +  0.08 x_{9} & -1.55 x_{2} & +  0.19 x_{10} & -1.13 x_{5} & -3.12 x_{3} & -0.12 x_{12} & +  2.08 x_{8}\\
 x_{13}   &  19.7096774194 & -0.40 x_{11} & -0.37 x_{9} & +  0.32 x_{2} & -0.29 x_{10} & -5.81 x_{5} & -4.90 x_{3} & +  0.10 x_{12} & +  1.72 x_{8}\\
\hline
z    &  86.6666666667 & +  5.00 x_{11} & +  0.22 x_{9} & -19.78 x_{2} & +  1.56 x_{10} & -18.78 x_{5} & -38.63 x_{3} & -1.41 x_{12} & + 26.07 x_{8}\\
\end{array}\]


 $ x_{8} $ enters and Unbounded Dictionary!


 $ x_{8} $ enters and Unbounded Dictionary!
\end{document}
